**E. Corbacho, V. Tarieladze, R. Vidal**

## Equicontinuity and
Quasi-Uniformities

**Abstract:**

For topological spaces $X,Y$ with a fixed compatible quasi-uniformity $\mathcal
Q$ in $Y$ and for a family $(f_i)_{i\in I}$ of mappings from $X$ to $Y$, the
notions of even continuity in the sense of Kelley, topological equicontinuity in
the sense of Royden and $\mathcal Q$-equicontinuity (i.e., equicontinuity with
respect to the topology of $X$ and $\mathcal Q$) are compared. It is shown that
$\mathcal Q$-equicontinuity implies even continuity, and if $\mathcal Q$ is
locally symmetric, it implies topological equicontinuity too. It turns out that
these notions are equivalent provided $\mathcal Q$ is a uniformity compatible
with a compact topology, but the equivalence may fail even for a locally
symmetric quasi-uniformity $\mathcal Q$ compatible with a compact metrizable
topology.

**Keywords:**

Topological space, quasi-uniform space, even continuity, topological
equicontinuity, uniform equicontinuity.

**MSC 2000:** Primary: 54C35; secondary: 54E15